Problem: Evaluate the iterated integral. $ \int_{3}^5 \left( \int_{2}^4 ye^{4x} \, dy \right) dx =$ Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{3}{2} (e^{20} - e^{12})$ (Choice B) B $\dfrac{3}{2}(e^{12} - e^{20})$ (Choice C) C $\dfrac{5}{2}(e^{20} - e^{12})$ (Choice D) D $\dfrac{5}{2}(e^{12} - e^{20})$
Solution: Evaluate the inner integral: $\begin{aligned} \int_3^5 \left( \int_2^4 ye^{4x} \, dy \right) dx &= \int_3^5 \left[ \dfrac{y^2e^{4x}}{2} \right]_2^4 dx \\ \\ &= \int_3^5 e^{4x} \left( \dfrac{16}{2} - \dfrac{4}{2} \right) dx \\ \\ &= \int_3^5 6e^{4x} dx \end{aligned}$ Evaluate the outer integral: $\begin{aligned} \int_3^5 6e^{4x} dx &= \dfrac{6e^{4x}}{4} \bigg|_3^5 \\ \\ &= \dfrac{3}{2} (e^{20} - e^{12}) \end{aligned}$ The answer: $ \int_{3}^5 \left( \int_{2}^4 ye^{4x} \, dy \right) dx = \dfrac{3}{2} (e^{20} - e^{12})$